SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Let  $(G, c)$  be a delegation graph and  $s \in S$  a sink. Let  $(\hat{G}, c)$  be the delegation graph, where the in-edges of  $s$  and all voters disconnected from casting voters are removed. Let  $P^{(\varepsilon)}$  and  $\hat{P}^{(\varepsilon)}$  be the transition matrices of the corresponding Markov chains  $M_{\varepsilon}$  and  $\hat{M}_{\varepsilon}$ . Then, for any  $\varepsilon > 0$  and edge  $e$  in  $\hat{G}$  we have  $P_{e}^{(\varepsilon)} \leq \hat{P}_{e}^{(\varepsilon)}$ . Since no edge on a path from any  $v \in N$  to any  $s' \in S \setminus \{s\}$  was removed, we have  $\hat{\mathcal{W}}[v, s'] = \mathcal{W}[v, s']$  and  $\hat{P}_{e}^{(\varepsilon)} \geq P_{e}^{(\varepsilon)}$  for every edge  $e$  in  $\hat{G}$  and  $\varepsilon > 0$ . Therefore, for the absolute voting weight of any  $s' \in S \setminus \{s\}$  in  $\hat{G}$  we get
$$
\pi_ {s ^ {\prime}} (\hat {A}) = 1 + \sum_ {v \in N} \lim  _ {\varepsilon \rightarrow 0} \sum_ {\hat {W} \in \hat {\mathcal {W}} [ v, s ^ {\prime} ]} \prod_ {e \in \hat {W}} P _ {e} ^ {(\varepsilon)} \geq 1 + \sum_ {v \in N} \lim  _ {\varepsilon \rightarrow 0} \sum_ {W \in \mathcal {W} [ v, s ^ {\prime} ]} \prod_ {e \in W} P _ {e} ^ {(\varepsilon)} = \pi_ {s ^ {\prime}} (A) ,
$$
Using Lemma C.1 and the proof of Theorem 7, we can show that guru-participation is satisfied by the RANDOM WALK RULE by removing a delegating voter step by step.
Theorem C.2. The RANDOM WALK RULE satisfies guru participation.
Proof. Let  $(G, c)$  be a delegation graph and  $v \in N$  a delegating voter. We remove  $v$  from  $G$  in three steps. First, we remove all out-edges of  $v$ , making  $v$  a casting voter and call the new delegation graph  $(\hat{G}_1, c)$ . Then we remove the in-edges of  $v$  (and all newly isolated voters) and get  $(\hat{G}_2, c)$ . Finally, we remove  $v$  itself to retrieve  $(\hat{G}, c)$  as in the definition of guru-participation. Let  $A$ ,  $\hat{A}_1$ ,  $\hat{A}_2$  and  $\hat{A}$  be the assignments returned by the RANDOM WALK RULE for  $(g, c)$ ,  $(\hat{G}_1, c)$ ,  $(\hat{G}_2, c)$  and
15More specifically, Brill et al. [2022] use the notion of relative voting weight between the casting voters in the definition of the axiom, which follows from our version of the axiom using absolute voting weight.
$(\hat{G}, c)$ , respectively. From the proof of Theorem 7 we know that for every casting voter  $s \in S \setminus S_v$  the voting weight in the instances  $(G, c)$  and  $(\hat{G}_1, c)$  is equal, i.e.,  $\pi_s(\hat{A}_1) = \pi_s(A)$ . From Lemma C.1 it follows that the voting weight of these voters can only increase if also the in-edges of  $v$  are removed, i.e.,  $\pi_s(\hat{A}_2) \geq \pi_s(\hat{A}_1)$ . Finally, removing the now completely isolated (now casting) voter  $v$  does not change the absolute voting weight of any other voter and therefore  $\pi_s(\hat{A}) \geq \pi_s(A)$ .
C.1 Relation to the Axioms of Brill et al. [2022]
First, we remark that the definition of a non-fractional delegation rule varies slightly from the definition of a delegation rule in Brill et al. [2022]. That is, Brill et al. [2022] define the output of a delegation rule as a mapping from each delegating voter to some path to a casting voter. Here, on the other hand, we define the output of a non-fractional delegation rule as a (non-fractional) assignment of delegating voters to casting voters. Hence, the definition of a delegation rule by Brill et al. [2022] is slightly more restrictive than our definition, hence, ceteris paribus, the impossibility result holds in particular for the smaller set of delegation rules. In the following, we refer to our definition as non-fractional delegation rules and to the definition of Brill et al. [2022] as non-fractional* delegation rules. We say that a non-fractional delegation rule is consistent to a non-fractional* delegation rule if, for any input, the assignment in the former corresponds to the induced assignment in the latter.
Copy-robustness Next, consider the copy-robustness axioms. The axiom by Brill et al. [2022] for non-fractional* delegation rules differs from our copy-robustness axiom restricted to non-fractional delegation rules in two technicalities: First, Brill et al. [2022] consider the relative voting weight instead of the absolute voting weight. However, since the number of voters does not change from  $(G,c)$  to  $(\hat{G},c)$ , this does not change the axiom. The other difference is that their axiom requires that the delegating voter  $v$  under consideration has a direct path (in the output of the non-fractional* delegation rule) to its assigned casting voter. Since a non-fractional delegation rule only outputs an assignment and no path, we relaxed this assumption. Hence, our copy-robustness axiom is slightly stronger than the one presented by Brill et al. [2022]. Nevertheless, it is easy to see that also the weaker version of the axiom is necessarily violated within the proof of the impossibility theorem when we utilize the definition of a delegation rule by Brill et al. [2022].
Confluence Lastly, consider the confluence axiom. Brill et al. [2022] define their confluence axiom, which we denote by  $\text{confluence}^*$  as follows: A non-fractional* delegation rule satisfies  $\text{confluence}^*$  if, for every delegating voter  $v \in N$  exactly one outgoing edge of  $v$  appears within the union of paths returned by the delegation rule. In particular, this is equivalent to the fact that the union of the returned paths forms a branching in the delegation graph. We prove below that the two axioms are in fact equivalent (within the restricted domain of non-fractional delegation rules).
Proposition 1. A non-fractional* delegation rule satisfies confluence* if and only if there exists a consistent non-fractional delegation rule that satisfies confluence.
Proof. We start by proving the forward direction. Consider a non-fractional* delegation rule that satisfies confluence*. We define a consistent non-fractional delegation rule by simply returning the assignment induced by the returned paths instead of the paths. For showing that the rule satisfies confluence, we define the probability distributions  $f_v, v \in N$  by setting  $f_v(W) = 1$  if and only if  $W$  is the path returned for  $v$  by the delegation rule. All other probabilities are set to zero. Then, confluence* directly implies that the distributions  $f_v, v \in N$  witness confluence.
For the other direction, consider a non-fractional delegation rule satisfying confluence, and let  $f_v, v \in N$  be the probability distributions that witness this fact. Building upon that, we define a branching in  $G$  that is consistent with the outcome of the delegation rule. Interpreting this branching as the output of a non-fractional* delegation rule then proves the claim. We construct the branching as follows: We first set  $B = \emptyset$ . In the beginning, set all delegating voters to be "active", while all casting voters are "inactive". Now, pick some arbitrary active voter  $v \in N$  and consider some arbitrary walk  $W$  that obtains non-zero probability by the distribution  $f_v$ . Construct a path  $P$  from  $W$  by cutting the walk in the first appearance of some inactive voter, and then short cutting all remaining cycles (if existent). Now, add all edges in  $P$  to  $B$  and set all delegating voters on  $P$  to be "inactive". Note that, by confluence, for each newly inactive voter  $u \in N$  it holds that the casting voter assigned by the delegation rule corresponds to the sink node at the end of the unique maximal path in  $B$  starting
from  $u$ . We continue this process until all voters are inactive. As a result, we created a branching that is consistent with the original delegation rule, hence, there exists a consistent non-fractional* delegation rule that implicitly returns branchings, i.e., is confluent.
D Broader Impact
We are aware of the fact that any delegation rule, and in particular the one suggested in this paper, may be implemented in a liquid democracy system and could thereby have real world impact. In this paper, we chose the axiomatic method in order to evaluate the suggested rule in a principled way. While, with respect to the axioms considered in the literature so far, our delegation rule performs very well, we want to point out that this is the very first paper introducing fractional delegation rules for ranked delegations. In particular, there is a risk of some unforeseen disadvantages of the rule that could possibly be used for manipulations or lead to other negative societal effects. Therefore, we think that further theoretical and also empirical research is necessary before recommending our suggested delegation rule for (high-stake) real-world decision making.